d
(d) According to Stefan’s law
\(E = eA\sigma {T^4} \Rightarrow {E_1} = {e_1}A\sigma T_1^4\) and \({E_2} = {e_2}A\sigma T_2^4\)
\({E_1} = {E_2}\) \(\therefore \)\({e_1}T_1^4 = {e_2}T_2^4\)
\( \Rightarrow \)\({T_2} = {\left( {\frac{{{e_1}}}{{{e_2}}}T_1^4} \right)^{\frac{1}{4}}} = {\left( {\frac{1}{{81}} \times {{(5802)}^4}} \right)^{\frac{1}{4}}}\)\( \Rightarrow \)\({T_B} = 1934\;K\)
And, from Wein’s law \({\lambda _A} \times {T_A} = {\lambda _B} \times {T_B}\)
\( \Rightarrow \frac{{{\lambda _A}}}{{{\lambda _B}}} = \frac{{{T_B}}}{{{T_A}}}\)\( \Rightarrow \)\(\frac{{{\lambda _B} - {\lambda _A}}}{{{\lambda _B}}} = \frac{{{T_A} - {T_B}}}{{{T_A}}}\)
\( \Rightarrow \)\(\frac{1}{{{\lambda _B}}} = \frac{{5802 - 1934}}{{5802}} = \frac{{3968}}{{5802}} \Rightarrow {\lambda _B} = 1.5\;\mu m\)